Question

A uniform iron bar of length 7 m weighs 22.4 kg. How much does the similar bar of length 13 m weigh?

Answer 1

Hint: The term ‘uniform’ denotes that the density of the bar is the same throughout the bar. Linear density (weight per unit length of the bar) $=\dfrac{Total\,\,weight\,\,of\,\,the\,\,bar}{Length\,\,of\,\,the\,\,bar}$
Weight of bar with length $l$ $=$ Weight per unit length of the bar$\times \,length(l)$

Complete step by step answer:
We are given that a uniform iron bar of length weighs 22.4 kg. We are asked to find the weight of the similar bar having length equal to 13 m. Let us assume the bars to be cylindrical.

First, we have to find the weight per unit length of the bar.
We know that,
Weight per unit length of the bar $=\dfrac{Total\,\,weight\,\,of\,\,the\,\,bar}{Length\,\,of\,\,the\,\,bar}$
$\therefore $ Weight per unit length of the bar $=\dfrac{22.4\,kg}{7\,m}=3.2\,kg/m$
Hence, weight per unit length (linear density) of the iron bar is $3.2\,kg/m$.
We are asked to find the weight of a similar bar having length $l$ equal to 13 m.
From the equation of linear density we know that,
Weight of bar with length $l$ $=$ Weight per unit length of the bar $\times \,length(l)$
We have already calculated weight per unit length (linear density) to be$3.2\,kg/m$.
$\therefore $ Weight of bar with length 13 m$=13\,m\times 3.2\,kg/m$ $=41.6\,kg.$

Hence, the weight of the uniform iron bar of length 13 m is 41.6 kg.

Note: Alternate method:
The same question can be done in terms of the volume of the iron bar.
Assuming that the bar is cylindrical, volume of the bar$=\pi {{r}^{2}}l$ (where $r$ is the radius of the cylindrical bar and $l$ is the length of the bar)
We can see that volume of the bar is directly proportional to the length of the bar $l$. Also, since the bar is said to be uniform, weight is proportional to volume. Therefore, the weight of the bar is also proportional to the length of the bar. Length 7 m corresponds to 22.4 kg and 13 m corresponds to the unknown weight.
Hence, by proportionality,
Unknown weight$=\dfrac{13\times 22.4}{7}kg=41.6\,kg$